RESEARCH BLOG 10/27/03 I'll make a couple of more comments on the results of Kronheimer et. Summary: RESEARCH BLOG 10/27/03 I'll make a couple of more comments on the results of Kronheimer et. al.. Last blog, I discussed 4-manifolds constructed from 4 or fewer han- dles. For and introduction to handle decompositions of 4-manifolds, see Gompf and Stipicz's book. The trickiest ones to classify are manifolds with two 2-handles. One way to view this is by considering pushing one 2-handle above the other. The level set in between the 2-handles is a 3-manifold, which is obtained by surgery on a knot in S3 in two different ways, by adding a two handle to the 0-handle, or by adding the other 2-handle to the 4-handle. The manifold then is determined by two knots in S3 which have the same surgery. If one performs inte- gral n-surgery on a knot, the resulting manifold has homology Z/nZ. Thus, to obtain the same manifold, we must be performing the same surgery (up to orientation reversal) on the two knots. Now, the main result of Kronheimer et. al. is that if a knot has the same Dehn surgery as the unknot, then it must be the unknot. This means that if one of our handles is unknotted, then the other is also. There is still some ambiguity in forming the manifold, since we must glue the boundary Collections: Mathematics